The invention is in the field of well logging, in which measurements taken in boreholes are used in searching for and exploiting valuable underground resources such as oil and gas. It is particularly directed to a method and a system related to energy windows logs, such as logs of the gamma radiation detected in several energy windows. It is more particularly directed to a method and system for converting energy windows logs into logs of selected subsurface materials such as thorium, uranium and potassium. It is further directed to a method and system which, while converting energy windows logs into subsurface materials logs, corrects for the borehole size and corrects for and produces additional logs of the radiation emitters and absorbers in the borehole fluid. The invention relates to logging natural as well as induced gamma radiation, and to other types of logging in which energy window measurements can be taken, such as sonic logging.
In the example of natural gamma radiation logging, a tool capable of detecting gamma radiation in each respective one of several energy windows passes through a selected borehole interval and measures the gamma ray photons detected in the respective windows in each respective one of a succession of small (e.g., 6-inch) depth intervals. The rays are emitted in the radioactive decay of subsurface materials such as thorium, uranium and potassium, each of which emits with a characteristic energy spectrum. The tool output is converted to a log of the respective emitting materials.
The log of materials such as thorium, uranium and potassium is important in the search for and exploitation of underground resources because it is believed that these materials appear in nature with a discernible relationship to geology and rock morphology. As some nonlimiting examples, it is believed that the ratio of thorium to uranium can be used for determining the geochemical faces in sedimentary rocks, the uranium to potassium ratio can be used to estimate the source rock potential of argillaceous sediments and that the thorium to potassium ratio can be used for determining the mineralogical composition of the shale. Perhaps more importantly, it is believed that by using the thorium, uranium and potassium concentrations either individually or in combination it is possible to measure the presence, type, and volume of shale or clay in the formations surrounding the borehole, which is particularly important in the search for and exploitation of oil and gas deposits.
While it is known that thorium, uranium and potassium emit gamma rays with characteristic discrete energy spectra, it is also known that between their emission and their detection those gamma rays undergo various interactions with the formation, the borehole and the tool and that, consequently, their apparent energy spectra as detected are continuous and have a poor energy resolution. Moreover, the borehole often contains borehole fluid (mud) which includes gamma ray emitters such as potassium chloride and gamma ray absorbers such as barite which affect the count rates detected in the energy windows. Additional uncertainties are introduced by the fact that relatively few gamma ray photons can be detected in the respective energy windows at a given borehole depth because the tool must move through the borehole at a sufficiently high speed to allow drilling or production activities to resume as soon as possible, and by the fact that the tool response changes as a function of borehole size.
Some aspects of known gamma radiation well logging are discussed in Marrett, G. et al., "Shaly Sand Evaluation Using Gamma Ray Spectrometry, Applied to the North Sea Jurassic," Proc. SPWLA 17th Annual Logging Symposium, June 9-12, 1976, and Serra, O. et al., "Theory, Interpretation and Practical Applications of Natural Gamma Ray Spectroscopy," Proc. SPWLA 21st Annual Logging Symposium, July 8-11, 1980, and additional information can be found in Chevalier et al. U.S. Pat. No. 3,976,878 and Moran et al. U.S. Pat. No. 3,521,064. As discussed in the cited documents, all of which are hereby incorporated by reference herein, it is possible to convert the output of a natural gamma radiation logging tool having three or five energy windows into a log of thorium, uranium and potassium concentrations (Th,U,K), in essence by subjecting the tool output to a filter characterized by a 3.times.3 or 3.times.5 matrix which can be empirically derived--such as by passing the tool through a test borehole containing known concentrations of Th,U,K arranged to approximate the effect of homogeneous beds of infinite depth extent and recording the windows responses. If W designates the radiation detected in five energy windows at a given borehole depth level, i.e., W=[W1, W2, W3, W4, W5], and X designates the thorium, uranium and potassium concentrations at the same depth level, i.e., X=[Th,U,K], then the relationship between the windows measurements W and the concentrations X (when no environmental effects are present) can be described by: EQU W=HX+.epsilon. (1a)
where H is defined by a 5.times.3 tool sensitivity matrix which is unique to a given tool and can be empirically derived by passing the tool through a borehole containing known concentrations of Th,U,K in idealized beds, and .epsilon.=[.epsilon.1, .epsilon.2, . . . .epsilon.5] denotes the statistical errors which are due to the Poisson nature of gamma ray detection. What is of interest normally is the concentrations of Th,U,K as a function of the radiation detected in the windows, and therefore what is of interest is the relationship: EQU X=MW (2a)
where M is defined as a 3.times.5 matrix relating the concentrations of the three materials to the radiation detected in the five energy windows at a given depth level in the borehole. The matrix M is not the inverse of the matrix H because a nonsquare matrix does not have a direct inverse in the strict sense of the term, but M can be found through a least squares technique relating known concentrations Th,U,K to measured radiation in the five energy windows for given test conditions. Of course the matrix M need be found only once for a given logging tool. In the case of a particular tool (see, e.g.) the Ellis article cited below) the following empirically derived numerical matrices H and M can be used for a standard (8"-diameter, water-filled) borehole: ##EQU1##
In the known technique, a log of the Th,U,K concentrations is derived by evaluating the relationship (2a) at each depth level in the borehole. Because of the nature of the logging process this estimate of Th,U,K concentrations tends to be noisy, but can be improved by some averaging of the radiation detected in the respective windows over successive borehole depth levels. For example, in order to find the Th,U,K concentrations at a given depth level n in the borehole, the matrix M can be applied to the average of the radiation detected in the five windows for the depth level n and the preceding one, n-1. The average need not be an arithmetic one, and more of the current depth level can be used than the preceding one. More than two depth levels can be averaged, but there is a limit because the consequence of averaging is a loss of resolution in the direction along the borehole axis.
Additional aspects of gamma radiation well logging, especially a method and system for correcting the log of selected gamma radiation emitting subsurface materials for environmental errors introduced by borehole size and by the gamma ray emitters and absorbers found in the borehole fluid, are discussed in Ellis, D., "Correction of NGT Logs for the Presence of KCl and barite Muds", Proc. SPWLA 23rd Annual Conference, July 6-8, 1982 (NGT is a trademark of Schlumberger). As discussed in that document, which is hereby incorporated by reference herein, the gamma radiation emitting subsurface materials logs may be corrected for at least one of: (i) the gamma ray emitter potassium chloride (KCl) in the borehole fluid, and (ii) a gamma ray attenuator (absorber) in the borehole fluid, e.g. barite and/or hematite. Thus the matrix H seen in expression (1b) and the relationship W=HX+.epsilon. seen in (1a) may be modified by considering the measurement of window 1 to be affected in a certain manner not only by radiation from the formation gamma ray emitting materials, but also by radiation from the KCl emitter in the borehole fluid, and by absorption (attentuation) by a strong absorber in the same borehole fluid. The measurements in windows 2 and 3 are affected both by radiation from materials in the undisturbed formations and radiation from the KCl in the borehole fluid. The contribution to the measurement in a given window from the KCl in the borehole fluid grows both as a function of borehole size and KCl concentration for both centered and eccentered tools. The contribution W(KCl) to the measurement in a given energy window which is due to KCl in the mud can be represented in the general case as EQU W(KCl)=(KCl)a[1-e.sup.-b(r-c) ] (3)
where a, b and c are constants which can be derived empirically by tests with a given logging tool in test boreholes having known diameters (r, found by a caliper log) and containing mud with known concentrations of KCl. The following relationships govern the lowest three energy windows in the non-limiting example of a particular 5-window logging tool, where W(KCl) is the contribution to the measurement in an energy window due to the borehole fluid KCl, KCl is in percent concentration and r is in inches: ##EQU2##
The relationship between window measurements and Th,U,K concentrations (related by the logging tool sensitivity matrix H) can be corrected for the influence of KCl and B in the borehole fluid by accounting for the effect thereon on the lowest three energy windows, in a relationship described by the following expression: ##EQU3##
The five unknowns in expression (2c) are the borehole fluid KCl concentration, the B correction and the concentrations of the three materials Th,U,K in the formations surrounding the borehole; and there are five relationships from which to find them.
In view of the known techniques discussed above for gamma radiation well logging and the corrections thereto for borehole fluid absorbers and emitters, and borehole size, one aspect of the invention relates to improving the log of Th,U,K concentrations derived from the radiation measured in five energy windows, based on the recognition that the concentrations log can be filtered not in a fixed manner but adaptively--in accordance with changes with borehole depth in the detected radiation and an understanding of the nature of the logging process. More particularly, this aspect of the invention is based on the discovery that a technique which has at least some characteristic of Kalman filtering can be used in connection with gamma radiation well logging when the nature of the logging process is taken into account in accordance with this invention.
In particular, in accordance with an illustrative and nonlimiting example of the invention, the Th,U,K concentrations log is derived by estimating the concentrations for a given borehole depth level through modifying the concentrations estimate for a previous depth level by an amount determined through applying a filter (constructed for the given depth level) to a combination of: (i) the radiation detected in the five energy windows for the given depth level and (ii) an estimate for the radiation in the five energy windows derived by applying the tool sensitivity matrix to the concentrations estimate for the previous depth level. If the filtered estimate for the Th,U,K concentrations for the current depth level n in the borehole is designated by X(n) and the filtered estimate for a previous depth level is designated by X(n-1), the filter gain for the given depth level is designated by K(n), the radiation detected in the five energy windows for the depth level n is designated by W(n) and the sensitivity matrix characterizing the particular well logging tool is designated by H, then one exemplary process in accordance with the invention can be described by the expression: EQU X(n)=X(n-1)+K(n) [W(n)-HX(n-1)] (4)
In a first, simpler and nonlimiting example of the invention, the filter gain K(n) for a particular borehole depth level n is determined by the behavior of the total detected gamma radiation in the vicinity of the depth level n. If the total detected gamma radiation is stable and smoothly varying in the relevant borehole depth interval, then in this example the effect of the adaptive filter gain K(n) approaches that of averaging the energy windows over a substantial borehole depth interval. However, if the total detected gamma radiation is changing significantly in the relevant borehole depth interval, then the gain K(n) approaches the 3.times.5 empirically derived matrix M discussed above. Stated differently, the filter adapts such that the previous Th,U,K concentrations estimate tends to dominate when the true concentrations at the relevant depth in the borehole are likely to be constant, and the newly measured energy windows tend to dominate otherwise. In a particular example, the gain K(n) can be the matrix M weighted by a scalar K(n) which varies between a value approaching zero and a value approaching unity.
In a second, more exacting but again nonlimiting example, the invention is implemented in a process which again applies a filter in the manner discussed in connection with expression (4) but the filter gain K(n) for a given depth level n is determined by a 3.times.3 matrix S characterizing the statistical variations in the filtered estimates of the Th,U,K concentrations, a 3.times.3 matrix Q characterizing the relevant geological noise, the well logging tool sensitivity matrix H, the matrix H transposed to produce a matrix H', and a 5.times.5 matrix R characterizing the statistical variations in the radiation detected in the five windows (i.e. the noise .epsilon.). The manner in which said factors determine the filter gain K(n) in this second exemplary embodiment of the invention can be described by: EQU K(n)=(S(n-1)+Q(n-1))H'[H[S(n-1)+Q(n-1)]H'+R(n)].sup.-1 ( 5)
where S is an estimate of statistical variations in the Th,U,K concentration estimates, Q is an estimate of the relevant geological noise and R is an estimate of the statistical variations in the energy window measurements.
In a third embodiment, where environmental effects are present, expression (1a) is modified to take into account the borehole size effect as well as the effect due to the presence of absorbers (e.g. barite) and radioactive emitters (e.g. KCl) in the borehole fluid. In a non-limiting example, where only barite and KCl are present in the borehole fluid, and where the borehole size effect has been modeled according to expression 8 below, expression (1a) is modified as follows: EQU W=H(x)X+.epsilon. (6)
where H(x) is a corrected sensitivity matrix function of the borehole absorption coefficient x. In turn, EQU x=.rho.(r-r.sub.sonde) (7)
where .rho. is the density of the mud, r is the diameter of the borehole (measured from a caliper log) and r.sub.sonde is the diameter of the sonde. The functional dependence on x of the (i,j)-th element [i=1, 2 . . . , 5; j=1, 2, 3] of the matrix H is given by EQU H.sub.ij (x)=(.alpha..sub.ij +.beta..sub.ij e.sup.-k ij.sup.x)H.sub.ij ( 8)
where H.sub.ij is the (i,j)-th element of the standard matrix (1b), and .alpha..sub.ij, .beta..sub.ij and k.sub.ij are calibration constants which can be found empirically.
The influence of barite and KCl has been modeled by D. Ellis, as noted above. With KCl denoting the concentration of KCl in the borehole fluid and B denoting the barite coefficient (acting as a gamma ray absorber for the first window only), the tool response relationship can be written as: ##EQU4## where H.sub.B (x)= ##EQU5## and where f.sub.i (x) denotes the influence of 1% of KCl in the mud on the i-th window (there being no influence on windows 4 and 5). The functional dependence of f.sub.i (x) on x is given by EQU f.sub.i (x)=a.sub.i (1-e.sup.-b i.sup.x)
as seen by relationship (3) and its following discussion.
Those skilled in the art will recognize that the third embodiment method for correcting for environmental effects is just a suitable modification of the method of the second embodiment of the invention. Instead of estimating only X=(Th,U,K), the potassium-chloride (KCl) and barite (B) concentrations must also be estimated at each depth. If, for simplicity, it is assumed that KCl and B are constant along the selected borehole interval (the case where KCl and B are slowly varying quantities being but a straightforward variation), and Y is defined as the vector of the (Th,U,K,KCl and B) concentrations, the estimation of Y is performed using the extended Kalman filtering technique, which is well-known by those skilled in the art. The estimate Y(n) of Y at depth n is given by EQU Y(n)=Y(n-1)+K(n) [W(n)-G(n)Y(n-1)] (10)
where G(n) is the following matrix: ##EQU6## with B(n-1) being the estimate of B at depth n-1, x.sub.n being defined for depth n as the borehole absorption coefficient of relationship (7), and b=H.sub.11 (x.sub.n) Th (n -1)+H.sub.12 (x.sub.n) U(n-1)+H.sub.13 (x.sub.n) K(n-1) [Th(n-1), U(n-1) and K(n-1) being the (Th,U,K) estimates at depth (n-1)].
The gain K(n) is now seen to be a (5.times.5) matrix given by EQU K.sub.n =[S(n-1)+Q(n-1)]G'(n)[G(n)[S(n-1)+Q(n-1)]G'(n)+R(n)].sup.-1 ( 12)
where S is a (5.times.5) matrix which represents the covariance of the statistical variations in the Th,U,K,KCl and B estimates; Q is assumed to be of the following form ##EQU7## where Q.sub.t, Q.sub.u and Q.sub.k are the variances of the geological noises in Th, U,K; and R is an estimate of the statistical variations in the energy window measurements.
In both the second and third embodiments, the log made up of samples (X)n or (Y)n estimated through use of the filter gain referred to in connection with expressions (5) or (12) can be further filtered, in a fixed-lag filter process which takes into account borehole depth levels subsequent to the one currently being processed.